You've 'based' your opinion 'on' faulty information. = Your opinion is 'based on' faulty information. [=the information that you used to form your opinion is wrong]
The company's 'base' is in London. = The company's 'base of operations' is (in) London. [=the company's main offices are in London]
Although I disagreed with the book's theoretical 'base' [='basis, foundation'], I couldn't really find fault with its logic.
(..) We're uploading your cerebral matrix...now. I can read him. He is dreaming. I wouldn't call it dreams. It's Ultron's base consciousness. Informational noise. Soon... How soon? I'm not being pushy. We're imprinting a physical brain. There are no shortcuts. Even if you imagined... Aaah! He... (..)2015 Avengers: Age of Ultron
(..) - Shit! - Language! Jarvis, what's the view from upstairs? The central building is protected by some kind of energy shield. Strucker's technology is well beyond any other Hydra base we've taken. Loki's scepter must be here. Strucker couldn't mount this defense without it. At long last. (..)2015 Avengers: Age of Ultron